TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 269, Number 2, February 1982
GEOMETRY AND THE PETTIS INTEGRAL1 BY ROBERT F. GEITZ
Abstract. Convex sets involving the range of a vector-valued function are con- structed. These constructions provide a complete characterization of the bounded Pettis integrable functions.
In this paper we present a geometrical study of the Pettis integral. Our results show that there is a strong link, previously unsuspected, between the range of a vector-valued function and its properties of measurability and integrability. In the course of this paper we will characterize the functions that are equivalent to strongly measurable functions (Theorem 2.8), the bounded Pettis integrable func- tions on arbitrary finite measure spaces (Theorems 3.2 and 3.4), and the bounded Pettis integrable functions on perfect finite measure spaces (Theorem 4.6). Along the way we will develop new insights into the nature of weak measurability and the Pettis integral. The link between measure theory and geometry in Banach spaces is not new. J. A. Clarkson defined the notion of uniform convexity almost fifty years ago as a condition to insure that functions of bounded variation are differentiable almost everywhere and have Bochner integrable derivatives. Radon-Nikodym theorems for the Bochner integral have been a fruitful source of geometrical properties. Rieffel introduced dentability in 1967 in the course of proving a Radon-Nikodym theorem. Slices and exposed points have similar origins. We are approaching measure theory from a different viewpoint from these authors. Rather than a global condition on the range space, we are looking for conditions on the range of an individual function that will guarantee that function to be integrable or measura- ble. This is in the spirit of the Pettis Measurability Theorem, which characterizes the weakly measurable functions that are strongly measurable in terms of the essential separability of their ranges.
1. Terminology. Let (£2, 2, p) be a finite measure space and let A be a Banach space. A function /( ■) from ñ into A is weakly measurable if the scalar function **/(') is measurable for each x* in the dual space A*. The function/ is Pettis integrable if for each E in 2 there is an element of A, denoted }E f dp, that satisfies x*JEf dp = jE x*f dp for every x* in A*.
Received by the editors January 20, 1981. Parts of this paper were presented at the 786th meeting of the Society in Pittsburgh, Pennsylvania on May 16, 1981. 1980 Mathematics Subject Classification Primary 28B05, 46G10. Key words and phrases. Pettis integral, weak measurability, perfect measure, Banach space. 'This material constitutes a portion of the author's Ph.D. thesis from the University of Illinois under the direction of Professor J. J. Uhl, Jr. © 1982 American Mathematical Society 0002-9947/82/0000-0771/$04.50 535 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 536 R. F. GEITZ
Another form of measurability can be attributed to some vector-valued func- tions. If/ is almost everywhere the limit (in the norm topology of A) of a sequence of simple functions, then / is strongly measurable. The Pettis Measurability Theo- rem [9, Theorem 1.1] says that a function is strongly measurable if and only if it is weakly measurable and off a null set it has separable range. If / is strongly measurable and if /||/|| dp < oo then the Bochner integral of/exists as an element of A [1, Theorem 2, p. 45], and/is trivially Pettis integrable. The Bochner integral has received considerable study-see [1]. Functions / and g with values in A are said to be equivalent if x*f = x*g a.e. for every x* in A*. Functions that are equivalent to strongly measurable functions need not themselves be strongly measurable. Example 2.3 gives a function / for which x*f = 0 a.e. for each x* in A*, yet/is not strongly measurable. 2. The core. The principal tool in this section, the core, is related to Rieffel's notion of the essential range of a vector-valued function. After developing a few of the technical aspects of this tool, we use it to characterize the functions that are equivalent to strongly measurable functions. Definition 2.1. Let /: ß —>A and let ££l The core of / over E, denoted corj(E), is the subset of A given by the formula cor/F)= H ~cöf(E\A).
Our first theorem suggests that core is closely tied to the integrability of vector-valued functions.
Theorem 2.2. Iff is a Pettis integrable function from ñ into X, then for each set E in 2 corf(E)= coi -J— fjdp: B g E, p(B) > oj.
Proof. Both directions of this equality are valid precisely because they hold for scalar functions. First, let 73 be a subset of E of positive measure, and let A be a null set. It follows easily from the Hahn-Banach theorem that
—l— [ fdpGœf(B\A)Gœf(E\ A). p(B) JB Hence co{( p(B))~x jBf dp: B G E, p(B) > 0} c cor/F). For containment in the opposite direction, let x be in cory(E), and let ||x*|| < 1. For any number e > 0 choose a countable partition tt of E and a function a, constant on the sets in tt, such that the inequality |x*/(0 — «(01 < £/4 holds for all t in E. Note that if B is any set in tt with positive measure and if t G B then, we must have the inequality x*™-mLxVdi\ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GEOMETRY AND THE PETTIS INTEGRAL 537 we have \x*(x)- 2 \*V(/,)| < «A Finally, for each number i let B¡ be the set in it containing t,. Observe that *•<*>-2v^ i/*'* x*(x) > inf C*{pW//^:5C£'M(i?)>0 It now follows from the Hahn-Banach theorem that x E CO: {(p(B)Yxjjdp: B g E,p(B) > oj. This completes the proof. This theorem has several immediate consequences. For example, if / is weakly equivalent to a strongly measurable function g, then / has nonempty core. To see why, observe that inside any set of positive measure there is a subset B such that gXB is bounded and Bochner integrable. It follows that fxs is Pettis integrable, and coiy (73) ^ 0. Note that if/and g are weakly equivalent Pettis integrable functions, then they have the same core, since this implies jBfdp = JBg dp for each measurable set 73. A later theorem (Theorem 2.6) will show that this does not depend on the integrability of / and g; equivalent functions always have the same core. Before exploring more fully the properties of the core, we present two examples. The first of these is an easy example that illustrates the utility of the core. Example 2.3. A function with nonseparable range whose core consists of a single point. Let /2[0, 1] be the usual space of countably nonzero functions on [0, 1] that are square-summable, under the /2-norm. For each point t in [0, 1] define e, G m 1]by , v [1 if s = t, e'(s) = 10 n if ( s =£^f t. Define/ from [0, 1] into l2[0, 1] by f(t) = e, for each / in [0, 1]. This function has nonseparable range; in fact, the span of its range is dense in the nonseparable space l2[0, 1]. However, since (/2[0, 1])* = /2[0, 1], the function x*/is only counta- bly nonzero for each x* in A*. Hence x*f = 0 a.e. (with respect to Lebesgue measure,) and/is weakly equivalent to the zero function. The core of/over any set of positive measure thus contains only the zero element of /2[0, 1]. The second example is much more complicated. This is Phillips's example [10, Example 10.8] of a bounded weakly measurable function that is not Pettis integra- ble. Example 2.4. A bounded nonintegrable function with empty core. Let lx[0, 1] be the usual space of all bounded real-valued functions on [0, 1], equipped with the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 538 R. F. GEITZ supremum norm. Under the assumption of the continuum hypothesis, Sierpinski constructed a subset B of the unit square [0, 1] X [0, 1] with properties (1) for each number t0 in [0, 1], the set {j: (s, t0) g B} is countable and (2) for each number s0 in [0, 1], the set {/: (s0, t) $ B} is countable. Let (fi, 2, p) be the standard Lebesgue measure space on [0, 1], and define a function/from [0, 1] into IJO, 1] by [f(s)](t) = XB(s,O- Phillips has shown that/is weakly measurable. In fact, for each x* in /^[0, 1], x*f is almost everywhere constant. We now show that the core of / is empty; hence / is not Pettis integrable. For each number t let A, be the set A, = {s: (s, t) G B}. The properties of B imply that H(At) = 0 for each t. Observe that if s & A, then [f(s)](t) = 0. Hence, if x E co/(fi \ A,) then x(t) = 0. If x E corAjl), then we must have x(t) = 0 for every /; the only possible element of cor^fi) is the zero element of lx[0, 1]. However, if 2 \f(Si) is a finite convex sum, in which s¡ G fi for every i, then 12 y(*,)|-sup |2W*')|-i. Hence 0 E co/(fi), and/has an empty core. We now pass to the study of equivalence to strong measurability. We require the following elementary lemma, whose proof we omit. Lemma 2.5. Let a and ß be measurable scalar-valued functions, and let p(E) > 0. Ifa(t) < ß(t)for all t in E then there is a set E* G E such that p(E*) > 0 and sup a(E*) < inf ß(E*). We come now to an important result; Theorem 2.6 has many applications. Theorem 2.6. Let f and g be weakly measurable functions from fi into X. If x*f = x*g a.e. for all x* in A*, then corA[E) = corg(E) for every measurable set E. Conversely, if both f and g have nonempty cores and if cor¿E) = corg(E) for every E in 2 then x*f = x*g a.e. for all x* in A*. Proof. First suppose that x*f = x*g. a.e. for all x* in A*. Let x be any element of corA[E), and let A0 be any null set. It suffices to show that for each x* in A*, x*(x) > inf x*g(E \A0). To this end^fix x* and let Ax = A0 (J {t: x*f(t) ¥= x*g(t)}. Clearly p(Ax) = 0; hence x E co/(F \AX), and x*(x) > inf x*f(E\Ax) = infx*g(E \ Ax) > inf x*g(E \ A0). For the second statement, suppose there is a functional x* in A* such that the condition of x*f = x*g a.e. fails. We may assume without loss of generality that p[t:x*f(t) Theorem 2.6 points out the utility of the notion of the core: this notion equates functions that the linear functionals believe are essentially the same. A function can have a very large range and still be weakly equivalent to a very simple function, as Example 2.3 shows. Thanks to this theorem, we can study equivalence to strong measurability by studying the ranges of strongly measurable functions. As a first step in this direction, we state a lemma that is well known. Lemma 2.7. A function g: fi —»A is strongly measurable if and only if for each set E of positive measure and for each e > 0 there is a measurable set E' G E such that p(E') > 0 and diam[ g(E')] < e. An immediate consequence of Theorem 2.6 and this lemma is that if / is equivalent to a strongly measurable function then inside every set there is a subset over which the core of/has small diameter. We now show that the converse of this statement is also true. The next theorem completely characterizes equivalence to strong measurability. Theorem 2.8. A weakly measurable function f: fi —»A is equivalent to a strongly measurable function if and only if for each set E of positive measure and for each e > 0 there is a set E' c E such that p(E) > 0, cor^F) ^ 0, and diam[cory(F)] in contradiction to inequality (*), and thus for each x* in A* p[t:x*f(t)^x*g(t)] =0. Lemma 2.7 and Theorem 2.8 may be summarized in the following terminology. A weakly measurable function is strongly measurable if and only if it has locally small range, and is equivalent to a strongly measurable function if and only if has locally small core. As an illustration of this, for each Lebesgue measurable set E of positive measure the function / in Example 2.3 has diam[/(F)] = V2 , but cor^F) consists only of a single point. In a sense, the intersections that form the core pare the range of a function down to its bare essentials. A function can have a very complicated range but still be simple from a measure-theoretic point of view. The core allows us to see the range of such a function in its simplest state. Before leaving the core, we note the following connection between it and a well-known set function. Suppose /: fi —»A is strongly measurable and E is in 2. Rieffel has defined the essential range of / over F, rA[E), as the collection of all elements x in A such that for each e > 0 the set {/ E E: \\f(t) — x\\ < e} has positive measure. Our concept of core is closely related to this notion. A straight- forward calculation places essential range in a geometrical setting: If / is strongly measurable then rf(E)= Pi f(E\A). After noting the similarity between this and the definition of cor^F), the following theorem is not surprising. Theorem 2.9. If f: fi —»A is strongly measurable, then for every set E in 2 corf(E) = cor [/y(F)]. The proof is not difficult and is omitted. 3. Pettis integrability on finite measure spaces. In this section we present two characterizations of the Pettis integrable functions. These theorems give the only known necessary and sufficient conditions for Pettis integrability with respect to an arbitrary finite measure. Definition 3.1. Let/: fi —>X, let B c fi be a measurable set, and let w be a finite partition of 73 into measurable sets. We denote by 5(77, B) the closure of the convex sel~2,EfE„cof(E)p(E). Now suppose that /: fi —>A is Pettis integrable. If Ti E 2 and if w is a finite partition of B, then f Jdp= 2 if dp G 2 œf(E)p(E). JB e License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GEOMETRY AND THE PETTIS INTEGRAL 541 Theorem 3.2. A bounded weakly measurable function /: fi -^ A is Pettis integrable if and only if (~)„ S(tt, B) =£ 0 for each measurable set B. Proof. The remarks preceding the statement of the theorem show that ¡Bfdp G fi „ S(tt, 73)iff is Pettis integrable. For the converse, we will show that jBfdp is the only possible element of fi, S(tt, 73). Fix 73 in 2 and assume that x E f)„ S(tt, 73); we shall show that x*(x) = fB x*f dp for every x* in A*. Fix such an x* and let e > 0. Since the scalar-valued function x*f is bounded, it is the uniform limit of a sequence of simple functions. It follows that there is a finite partition tt of 73 such that if F is in tt and if /, and t2 are points in E, then \x*f(tx) - x*f(t2)\ < e. Hence for any point / in E, we have 1 [ x*fdp - x*f(t) f x*fdp- x*(xE)p(E) < ep(F). J F Summing over all the sets F in tt gives \fB x*f dp — x*(x)| < ep(B). Since this holds for each e > 0, we have x*(x) = /B x*fdp, and Pettisis- / / dp. JB One way to insure that the sets 5(77, 73) have nonvoid intersection is to require them to meet a weakly compact set. Corollary 3.3. A bounded weakly measurable function f: fi —»A is Pettis integra- ble if and only if for each set B in 2 there is a weakly compact set W G X such that W n 5(77, 73) =£■0 for every finite partition tt of B. Proof. If/ is Pettis integrable, let W be the weak closure of the set {fEf dp: E G 2}. This set is well known to be weakly compact (see [1, Corollary II.3.9]) and we have already seen that W n 5(77, B) i= 0 for every 77and B. To prove the opposite implication, observe that if 77 and 77' are partitions of 73 such that 77' refines 77, then 5(77', 77) c 5(t7, 77). Hence the family of sets { W n 5(77, 77): 77is a portion of 73} has the finite intersection property. Since W is weakly compact and these sets are weakly closed, this family has nonempty intersection. This holds for each B in 2; by Theorem 3.2 is Pettis integrable. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 542 R. F. GEITZ Our second characterization looks back to Theorem 2.2, where we showed that the average values of a Pettis integrable function lie in its core. We now show that the indefinite Pettis integral is the only vector measure F such that F(E) G co f(E)p(D) for every measurable set E. Theorem 3.4. A bounded weakly measurable function f: fi —»A is Pettis integrable if and only if there is a finitely additive vector measure F: 2 —»A such that F(E) G œ f(E)p(E) for every set E in 2. Proof. Suppose / is Pettis integrable and write F(E) = fEfdp for E in 2. Appeal to [1, Theorem II.3.5] to see that F is a countable additive vector measure. A glance at the introduction to this section shows that F(E) E co/(F)p(F). To prove the converse, note that for any set B in 2 and any finite partition 7r of B, F(73)= 2 F(F)E 2 œ/(F)p(F) c 5(77,73). E ŒlT EE.7T The conclusion now follows from Theorem 3.2. 4. Integration on perfect measure spaces. In this section we restrict our attention to integration on perfect measure spaces. This is a very broad class of measure spaces first defined by Gnedenko and Kolmogorov for the purpose of avoiding pathological behavior in probability theory. They were the subject of an extensive study by Sazonov [12] and form the background for a major work by Fremlin [6]. As we show in [7] and in the present work, they also eliminate much of the pathological behavior of the Pettis integral. Charles Stegall has shown [6, Proposition 3J] that the indefinite Pettis integral with respect to a perfect measure has relatively norm compact range. This strengthens one direction of Corollary 3.3. If (fi, 2, p) is perfect and if/: fi -» A is bounded and Pettis integrable, then let K be the norm closure of the set {jE F dp: E G 2}. Stegall's theorem shows that K is compact, and clearly K n 5(77, 73) ^ 0 for every partition 77 of 73. We thus have the following. Theorem 4.1. Let (fi, 2, p) be a perfect measure space, and let f: fi —*X be a weakly measurable function. Each of the following statements implies all the others. (l)f is Pettis integrable. (2) For each b in 2 there is a weakly compact set W c A such that W n 5(77, B) =^=0 for every partition tt of B. (3) For each B in 2 there is a norm compact set K G X such that K n 5(77, 77) ¥= 0 for every partition tt of B. In the remainder of this section we consider three set functions whose values are contained in the core of /. Since we are considering only bounded functions, without loss of generality we may assume ||/(r)|| < 1 for all t. The definitions below may be easily altered to encompass arbitrary bounded functions. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GEOMETRY AND THE PETTIS INTEGRAL 543 Definition 4.2. Let/be a function from fi into the unit ball U of A and let E be a measurable set. The sets Gf(E) and Hj(E) are defined by Gj(E)= H w[f(E\A)p(E\A) + p(A)U], AczE and Hj(E)= fi œ[f(E\A)p(E) + 3p(A)U], AdE where the intersections are taken over all measurable proper subsets of E. The set Qj(E) consists of those x in A such that the inequality p[t S E: x*f(t) < x*(A) + e] > ep(F)/2 holds whenever ||x*|| < 1 and 0 < e < 2. It is immediately clear that these sets are convex, that Ga[E) and Hj(E) are closed, that GA[E) c Hj(E) c cory(E), and that fEfdp G Ga[E) whenever / is Pettis integrable. We shall see later that G/F) c Qj(E)p(E) c 77/F), but that Qf(E) is not necessarily closed. These sets are therefore distinct. We are now going to show that when (fi, 2, p) is perfect these sets have two surprising properties. (1) for every F in 2 the sets Gj(E) and Hf(E) are compact, and (2) the function / is Pettis integrable whenever GA[E) and Ha[E) are nonempty for all sets E of positive measure. The first lemma for this is an analogue for the norm topology of a classic theorem of R. C. James. Lemma 4.3. Let C be a closed bounded set in X. If C is not norm compact then there is a number e > 0 and there are sequences (xk) in C and (x*) in the unit ball of X* such that x*(xn) > e and x*(xk) = 0 when k < n. Proof. Select a number e > 0 such that C cannot be covered by finitely many balls of radius 2e. Inductively select a sequence (xk) in C such that (*) dist(x„ + 1, span[x1; . . . , x„]) >e for all integers n as follows. Let Xj be any element of C. If x,, . . . , x„_, are elements of C that have been chosen to satisfy (*), then there must be an element of C not in the set spanfx,, . . . , x„_,] + eU, or else C could be covered by finitely many 2e balls. Let this point be x„, and let dn be the distance from x„ to span[x,, . . ., x„_,]; clearly dn > e. Observe that the sequence (xk) thus selected is linearly independent. For each integer n we define a functional \pn on span[x„ . . ., xj by «r-J 2 «,*,] = dnan for all real numbers a,, . . . , an. An easy calculation shows that || License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 544 R. F. GEITZ Theorem 4.4. Let (fi, 2, p) be a finite perfect measure space and let F E 2. Iff is a bounded weakly measurable function from fi into the unit ball of X then HÁE) is compact. Proof. If Ha[E) is not compact then by Lemma 4.3 we may select a sequence (xk) in Hj(E), a bounded sequence (x*) in A*, and a number e > 0 such that x*(xn) > e Ior all n, and x*(xk) = 0 for k < n. Applying [7, Lemma 1] to the sequence (x*f), we may extract a subsequence (x*f) and find a weak* cluster point x* of (x*) such that lim, x*f = x*/ a.e. Egorov's theorem produces a set A such that (x*f) converges uniformly on£\^ and p(A) < e/10. Let mx and m2 > mx be any two indices for this subsequence such that the inequality ko*/(0- „ - p(£)2 «At,)< e/10 + 3p(A) < 2e/5. Hence, for all x* in the unit ball of A*, (•) x*(xn) - p(E) 2 «,**/(',) < 2e/5. To produce a contradiction, we first let x* = x*^ Since |x* f(t¡) — x*/(f,)| < e/10p(F) holds for 1 < i < n, the inequality xL(xm) - n(E) 2 «í-*ó7('/) < e/10 + 2e/5 = e/2 holds. Since x* (xm ) > e, we see that |p(F)2"=1 cXjXJUfitj)]< e/2. On the other hand, letting x* = x*2 in (*) yields | p(F)2"=, a,x*J < 2e/5; hence p(F)2 OiXSÄtt) This is a contradiction and proves that Ha[E) must be compact. Lemma 4.5. 77yis superadditive, i.e., 77/F,) + 77/F2) c 77/F, u F2) whenever F, and E2 are disjoint measurable sets. Proof. Let x, E Ha[E¡) for / = 1,2, and let A c Ex u F2. We must show that x, + x2 E cô[/(F, U E2\A)p(Ex U F2) + 2>¡i(A)U~\. Case 1. If F, \ A =£0 and F2 \ A =£0 then xx + x2 G cö [ /(F, \ A)ii(Ex) + 3¡i(A n F,) U] + œ[f(E2 \ A)¡x(E2) + 3p(^l n E2)U] Cco[/(F, u E2\A)p(Ex u F2) + 3p(A)U]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GEOMETRY AND THE PETTIS INTEGRAL 545 Case 2. If F, c A then, since A does not contain both Ex and E2, there are elements in z in co f(E2 \ A) and u in U such that x2 = zp(E2) + 3¡x(A n E2)u = zp(Ex u F2) - zp(Ex) + 3p(A n F2)m. Also note thatx, E 77/F,) c p(Ex)U. Thus ||x, - zp(F,)|| < 2p(F,), and x, + x2 E co f(E2 \ A)p(Ex u F2) + 3p(Ex)U + 3p(E2 n A)U G cö [/(F, u F2 \ A)p(Ex u F2) + 3p(A)U]. The importance of the set function Hf will now become apparent. Theorem 4.6. Let (fi, 2, p) be perfect. A bounded weakly measurable function f from fi into the unit ball of X is Pettis integrable if and only i/77/F) =£ 0 for every set E of positive measure. Proof. One direction of this is immediate: iff is Pettis integrable then ¡Efdp G 77/F). For the converse we apply Corollary 3.3. Let 73 be any set of positive measure. If 77 is a partition of 77, then 2£ew 77/F) c 2£e^ co f(E)p(E) c 5(t7, B). The superadditivity of Hf shows that 2£e,r /3/F) c Hj(B). Hence 77/77) n 5(tt, 73) =£ 0 for all partitions 77of B, and / is Pettis integrable. Of course, since G/F) c Ha[E), Theorems 4.4 and 4.6 are valid with 77/F) replaced by Ga[E). We turn now to the set function Ö/F). Theorem 4.7. 7/p(F) > 0, then G/F) c Qj(E)p(E) G 77/F). Proof. First we show that Gj(E) c Q/(E)p(E). Let x be in Ga[E). For any functional x* in the unit ball of A* and for any positive number e < 2, let A = {t G E: x*f(t) < x*(x)/p(E) + e}. Observe that x E co[/(F \ A)p(E \ A) + p(A)U]; hence inf[x*f(E \A)]p(E\A)< x*(x) + p(A), and thus [x*(x)/p(E) + e]p(E \A) < x*(x) + p(A). Solving this inequality for p(A) gives 1 a\ ^(E) M( ' > 1 + x*(x)/p(E) + e ' Observe that if x*(x)/p(F) + e > 1 then A = E, and the conclusion p(A) > ep(E)/2 holds trivially. On the other hand, if x*(x)/p(E) + e < 1, then again p(A) > ep(E)/2. In either case p[tGE: x*f(t) < x*(x)/p(E) + e]> ep(E)/2, and x/p(E) G Ö/F). Now let x be in £?/F); it is apparent that ||x*|| < 1. Let A c F. If p(A) = p(E), then x E U G~c~3[f(E \A) + (3p(A)/p(E))U]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 546 R. F. GEITZ If p(A) < p(E), then for each x* in the unit ball of A* the inequality p[t G E: x*f(t) < x*(x) + 2p(A)/p(E)] > 11(A) holds. Hence there is a point t in F \ A such that x*f(t) < x*(x) + 2p(A)/p(E), and thus x*(x) > inf[x*/(F\^)] - 2p(A)/p(E). By the Hahn-Banach theorem, we have xp(E) G~œ[f(E\A)p(E) + 2p(A)U] c co [f(E \ A)p(E) + 3p(A)U]. Thus xfx(E) G 77/F), and this completes the proof. 5. Examples. In this section we present several examples of Pettis integrable functions and the set functions associated with them. The first example shows that even when /is real-valued, the set g/F) need not be closed. Example 5.1. A function for which Q/\E) is not closed. Let p represent Lebesgue measure on fi = [0, 1], and let/: fi —>A be defined by f(t) =Vt . Straightforward calculations show that G/fi) =[5/27, 23/27], g/fi) =(1/8, 1], 77/fi) =[ 1/12, 1], and cor/fi) = [0, 1]. Notice that these sets are increasing and that each contains faf dp = 2/3, as required by our work in the preceding section. Example 5.2. Let (F„) be a sequence of disjoint sets covering fi = [0, 1], such that p(F„) = 2~" for every integer n. Let (en) be the unit vector basis of /, and define/: fi —»/, by At) = 2 Xe„(t)en. n By Theorem 2.2, we know cor/fi) = co{/& f dp/fi(En)} =co{e„}. In other words, cor/fi) consists of those elements of the unit sphere of /, with nonnegative entries. If x = (x„) is in G/fi), then for each integer n there must be nonnegative scalars (a¡) such that 2,^„ a, = 1 and x E (2,^ a,.e,.)(l - 2'") + 2~nU. Hence x„ < 2~" for each n. Since G/fi) c cor/fi), we see that the only element of G/fi) is 2 2~\ = f dp. n JQ. In a similar manner, it is quite easily seen that x = (x„) is in 77/fi) if and only if 0 < x„ < 3/2" and 2„ x„ = 1. Now consider g/fi). By letting x* = -c* E /„ and requiring u[t: x*f(t) < x*(x) + e] > e/2, we find necessary conditions for g/fi) similar to those for 77/fi). If x = (x„) is in g/fi), then 0 < x„ < 2/2" and 2„ x„ = 1. These conditions, however, are not sufficient; the element ex is not in g^fi), as we see by letting x* be the sequence <-l, 1, 1, 1, ... > and letting e = 3/2. Example 5.3. Consider the function defined by fit) = X[o,i]G Ex. To calculate the integral off, let \p be in Lx, and let x* be the element of L^ corresponding to \p. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GEOMETRY AND THE PETTIS INTEGRAL 547 For any number b > 0, we have [" x*f(t) dp(t) = [" V *(j) dp(s) dp(t) = f" f Ms) dp(t) dp(s) = f xp(s)(b- s) dp(s). •'0 Js J0 Hence jçf(t)dp(t) = (b — s)X[oyb](s)and the integral is the element of ¿M with graph Note also that fbaf(t)dp(t) b- s b - a = X[0,a)(i) + b _ Q X[a, b)(s), and the average value of / over the interval [a, b] is thus the element of Lx with graph For each interval [a, b], the average value of / over [a, b] is an element of La with the properties (i) hence every element of cor/fi) must satisfy (l)-(3). On the other hand, any function satisfying (l)-(3) can be uniformly approximated by piecewise-linear functions in cor/fi). The core of/over [0, 1] therefore consists of all the elements of Lx that satisfy conditions (l)-(3). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 548 R. F. GEITZ Now let Bibliography 1. J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R. I., 1977. 2. N. Dunford and J. T. Schwartz, Linear operators. I, Interscience, New York, 1958. 3. G. A. Edgar, Measurability in a Banach space, Indiana Univ. Math. J. 26 (1977), 663-677. 4._, Measurability in a Banach space. II, Indiana Univ. Math. J. (to appear). 5. D. H. Fremlin, Pointwise compact sets of measurable functions, Manuscripta Math. 15 (1975), 219-242. 6. D. M. Fremlin and M. Talagrand, A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means, Math. Z. 168 (1979), 117-142. 7. R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. (to appear). 8. R. C. James, Weak compactness and reflexivity, Israel J. Math. 2 (1964), 101-119. 9. B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. 10. R. S. Phillips, Integration in a convex linear topological space, Trans. Amer. Math. Soc. 47 (1940), 114-145. 11. M. A. Rieffel, The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968),466-487. 12. V. V. Sazonov, On perfect measures, Amer. Math. Soc. Transi. (2) 48 (1965), 229-254. Department of Mathematics, Oberlin College, Oberltn, Ohio 44074 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use